Wythoff’s Game

This is a quick game for two players. Take an empty chessboard and place a queen on h5.  The players take turns moving the queen (both players move the same piece) either left, down, or diagonally down and left (indicated by the arrows in the diagram). The first person to move the queen to a1 (the bottom left hand corner shown with a flag).

Fig.1 One queen Wythoff’s Game

 

 

 

 

 

 

 

 

 

The way to analyse this game is to classify each square as being safe or unsafe. If you can get to a safe square you can win. The real work comes beforehand in drawing a map of safe and unsafe squares. As a classroom exercise, coloured counters can be used to mark the board.

We work backwards, step by step. You need to avoid placing a queen in line of sight of the flag. These squares are taboo and marked in red (Fig. 2).

Fig. 2 Green = safe; Red = unsafe

 

 

 

 

 

 

 

 

 

The green squares on b3 and c2 are safe squares because the next player can only move to a square which is in line of sight to the flag.  For the next step, we continue to work backwards and mark as unsafe (in purple) those squares which have line of sight to this pair of safe squares.

Fig 3. Three safe squares in Green

 

 

 

 

 

 

 

 

 

We now find that there is another safe square on f4, also marked in green. For the third step we mark as unsafe all those squares which are in line of sight of f4. These are shown with red crosses.

Fig. 4 The White queen cannot reach a safe square

 

 

 

 

 

 

 

 

 

The first player cannot get to the safe square f4. It can only move to a square which is red, purple or marked with a red cross.  This implies that whichever move the first player makes, the second player can find a safe square and will win the game.

Extensions

You can also play simpler versions of the game with a king or a rook subject to the constraint that moves can only be towards the bottom left hand corner.

The complexity of the game can be increased by introducing a twin piece. In addition to the queen on h5 another queen is placed on g8. Players can choose which of the two queens to move with the objective of being the first to reach the flag.

Two queen – Wythoff’s Game

 

 

 

 

 

 

 

 

 

What makes this version more challenging is that there are so many combinations of moves for the two queens which can also block each other.